For a an element of U\N U (K), we let μ a be the normalized singular measure supported in KaK.

THE HAAR MEASURE 3 2. In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. Haar measure on SU(2) 2.1.

2.2 HAAR MEASURE AND FUNCTION SPACES ON SU(2) ..... 31 2.2.1 The Haar Measure on SU(2) and its Properties.....31 2.2.2 Integration and Convolution on L. 1 (SU(2)) ..... 32 2.2.3 The Q. x. Haar measure on a locally compact quantum group Byung-Jay Kahng Department of Mathematics, University of Kansas, Lawrence, KS 66045 e-mail: bjkahng@math.ku.edu Communicated by: V. Kumar Murty Received: July 15, 2003 Abstract. The goal is to explicitly describe a Haar measure on SU(2) with respect to which we can define and then discuss properties of the integral. Haar Measure on E q(2) Arupkumar Pal Indian Statistical Institute 7, SJSS Marg, New Delhi 110016, INDIA e-mail: arup@isid.ernet.in The quantum E(2) group is one of the simplest known examples so far of a locally compact noncompact quantum group.

The goal. Thus, describing a Haar measure on SU(2) reduces to defining a Haar measure on S3. In the general theory of locally compact quantum groups, the notion of Haar measure (Haar weight) plays the most significant role. The Haar measure is, by definition, the unique group-invariant measure, so it is used to average properties that are not unitarily invariant over all states, or over all unitaries. The aim of this paper is to … THE HAAR MEASURE OF A LIE GROUP a simple construction L.Molinari G is representation of a Lie Group, with elements U that are unitary matrices of size N. In the exponential form U = eiH, the Hermitian N × N matrix H belongs to a Lie algebra. For this system the relevant group is SU(2) which is the group of all 2x2 unitary operators. A particularly widely used example of this is the spin system. The existence and uniqueness of an ‘invariant measure’ on this group has been proved in this article. We let U=SU(2), K=SO(2) and denote by N U (K) the normalizer of K in U. Definition and examples of Haar measure Let Gbe a locally compact group, and let be a Borel measure on G, with Bthe Borel ˙-algebra of G. The measure is left translation invariant if for every S 2B, (gS) = (S) for all g2G, and is right invariant if for every S2B, (Sg) = (S) for all g2G. Operator on L. 1 (SU(2)) and its Properties.....37 2.2.4 Di erential Operators on SU(2).....44 2.3 REPRESENTTIONA THEORY ON SU(2).....53 2.3.1 The Subspaces M. m. and M~ m.....53 2.3.2 …

Including the Haar measure we show that the effective potential of the regularized SU(2) Yang-Mills theory has a minimum at vanishing Wilson-line W = 0 for strong coupling, whereas it develops two degenerate minima close to W = ±1 for weak coupling.